Gaslight and Cavell.
Cavell drives the parallel between Descartes, who is doubting reality and begins with “original foundations” [CP, 13] for sake of pure reasoning and Paula, who's questioning own madness. Descartes reject his opinions merely because he found reasons to doubt them [senses, dreams and evil demon]. For Paula’s process starts only at the end of the movie, when she sees her tied husband in the attic; she sings a “mad song” [CT, 76] to him. Her experience of reality was constrained by detective [who quite pressured her into it] and the true about her husband. In both cases we can question the “purity” of experience, because it isn’t very clear weather both, Descartes and Paula, are free of own bias. Descartes makes an assumption of necessity of supreme being, which is already somewhat bias and Paula’s condition before and after marriage isn’t pronounced in the movie, it rather stays unclear after all.
Paula's memories of the past were disfigured by her husband, in attempt to drive her crazy. Accepting the reality becomes a way of gaining certainity of own existance. Descartes gains one due to internal impulse that caused him to build a thought experiment, which led him to acknowledge own existance trough recognizing own ability to think, which in its turn is obligated to have a source -- Decartes himself. Paula gains her experience of reality with help of “external irritant” (detective and impossibility to explain to herself all husband’s misconducts) and gains certain freedom as her “aria of revenge” [CT, 70] goes on. The moment when she pronounced her thoughts (I'm mad, therefore I am?) the self-identification begins. More than that, according to Cavell's notion of Emersonian prove of existance, she exposes the counchesness to others [CT, 71-72].
Paula, who used to explain herself all the strange things with help from her husband, finally achieves counchesness that makes her exhistance real. In her case, accepting own madness and proving it to the others, to certain extend, is the price she pays in terms to achieve own cogitatio and become independent from her Gregory's projections. She gaining not only rational, but also emotional freedom from tyrannical husband. Losing the marrige for her (as much as for viewer) more an award than punishment. Whether she had her Cogito prior marrige or not, 'I am mad’ (which Stanley Cavell takes to reveal her Cogito at the very end of the movie), leaves Ingrid Bergman’s character at the new stage of experiencing, called “I am thinking thing”.